Gaussian Process (GP) models are often used as mathematical approximations of time expensive numerical simulators. Provided that its kernel is suitably chosen and that enough data is available to obtain a reasonable t of the simulator, a GP model can be benecially used for as many tasks as prediction, optimization, or Monte-Carlo-based quantication of uncertainty. However, the former conditions become unrealistic when using classical GPs as the dimension of input increases. One popular alternative is then to turn to Generalized Additive Models (GAMs), relying on the assumption that the simulator's response can approximately be decomposed as a sum of univariate functions. If such an approach has been successfully applied in approximation, it is nevertheless not completely compatible with the GP framework and its versatile applications. The ambition of the present work is to give an insight into the use of GPs for GAMs by integrating additivity within the kernel, and proposing a parsimonious numerical method for data-driven parameter estimation. The first part of this paper deals with the kernels naturally associated to additive processes and the properties of the GP models based on such kernels. The second part is dedicated to a numerical procedure based on relaxation for additive kernel parameter estimation. Finally, the efficiency of the proposed method is illustrated and compared to other approaches on Sobol's g-function in dimension 4, 8 and 12.